Exploring complexity of class-A Beta-lactamase family using physiochemical-based multiplex networks

The Beta-lactamase protein family is vital in countering Beta-lactam antibiotics, a widely used antimicrobial. To enhance our understanding of this family, we adopted a novel approach employing a multiplex network representation of its multiple sequence alignment. Each network layer, derived from the physiochemical properties of amino acids, unveils distinct insights into the intricate interactions among nodes, thereby enabling the identification of key motifs. Nodes with identical property signs tend to aggregate, providing evidence of the presence of consequential functional and evolutionary constraints shaping the Beta-lactamase family. We further investigate the distribution of evolutionary links across various layers. We observe that polarity manifests the highest number of unique links at lower thresholds, followed by hydrophobicity and polarizability, wherein hydrophobicity exerts dominance at higher thresholds. Further, the combinations of polarizability and volume, exhibit multiple simultaneous connections at all thresholds. The combination of hydrophobicity, polarizability, and volume uncovers shared links exclusive to these layers, implying substantial evolutionary impacts that may have functional or structural implications. By assessing the multi-degree of nodes, we unveil the hierarchical influence of properties at each position, identifying crucial properties responsible for the protein’s functionality and providing valuable insights into potential targets for modulating enzymatic activity.


Weighted multiplex threshold network
For each physiochemical property, denoted as α , we calculate the Pearson's correlation coefficient between dif- ferent positions of the data matrix D α .The correlation coefficient C α ij between position i and j of the data matrix D α for a fixed α is given by Here, d α i represents the i th column of the data matrix D α , σ (d α i ) represents the standard deviation of i th column of the data matrix D α and < • • • > denotes the average over the proteins sequences 13 .These correlation coeffi- cients are utilized to construct a network that represents the evolutionary interaction for the class-A β-lactamase enzyme family.However, due to the multiple relationships among the nodes (positions) in the network resulting (1) . from different physiochemical properties, a simple network representation is insufficient.To capture the multiple interactions among the nodes effectively, we employ a multiplex network approach 29 .A multiplex network consists of layers, where the same set of nodes exists in different layers and each layer represents a different form of interaction or communication among entities.
A multiplex network is a set of networks G α = (N, E α ) arranged in layers, with α = 1, . . ., L with L as the number of layers.The set of nodes N is the same in each layer whereas the set of edges E α is layer dependent.For the class-A β-lactamase enzyme family, each multiplex network layer G α (N, E α ) , is characterized by α = 1, . . ., L corresponds to a specific physiochemical property, the nodes (N) represents the positions in the MSA and a edge E α i,j between two nodes i and j is given by where θ is the threshold value and C α ij is the Pearson correlation coefficient between position i and j for a physiochemical property ( α ).The threshold serves as a cutoff value for deciding which correlations are significant enough to be represented in the threshold network.Correlation coefficients that meet or exceed this threshold value are considered strong correlations and are included in the network, while those below the threshold are excluded.In the multiplex network, the interlayer links connect a given node to its counterpart nodes in the rest of the layers.The intra-layer links physically represent the extent of the evolutionary interaction between two positions i and j in a layer.In present context, the evolutionary interaction refers to the co-evolutionary relationships among amino acids physicochemical properties within the class-A β-lactamase enzyme family that are conserved throughout the evolutionary history of the protein.We aim to comprehend how these physiochemical properties have collectively evolved over time, providing insights into their functional and structural roles.The links that are preserved at higher thresholds indicate the interactions that have been conserved throughout the evolutionary history of the protein.A link is present between two positions if the absolute correlation strength between them is greater than the threshold for the given physiochemical property.Since 4 different physiochemical properties are used, the multiplex network has 4 different layers ( L = 4 ), each layer representing one property.The complete multiple network is denoted as , where G α is the network corresponding to the property α .Each physiochemical property contributes a network layer with the same set of nodes but with a different set of connections.The changes in the network with the number of sequences used for analysis is given in the Supplementary Material Tables 1 and 2.
By changing the threshold, the multiplex network structure changes.An increase in threshold results in the isolation of nodes within the network layers whose interaction strength with other nodes is smaller than the threshold value for the given physiochemical property.Higher thresholds lead to sparser networks with stronger interactions among nodes within each layer.Figure 2 shows the different network layers at threshold values of 0.3, 0.5, and 0.7, respectively, with only nodes having non-zero connections shown for clarity.The nodes are colored by the average value of the physiochemical property at that position.Mathematically, the average value at position i for property α is given by d α i where d α i represents the i th column of the data matrix D α and • • • is the average over the sequences.If the average value of a physiochemical property α at a position is positive then the node is colored red and a blue-colored node implies a negative value of the physiochemical value at that position.By analyzing each layer of the network, it is evident that the positions tend to interact and connect with the other positions having similar values of the physiochemical properties.We observe the formation of patterns in each layer based on the average value of physiochemical properties, where for each layer the blue nodes tend to have higher connections with blue nodes whereas red nodes tend to connect with red nodes.At higher thresholds (Figure 2), the nodes are completely separated into components based on the average value of the physiochemical property.This implies that the interaction between positions depends highly on the value of the physiochemical property.There are only a few interactions between the two groups (red and blue) of node, most of the interactions are among the members of the group.

Filtering edges, information loss, and randomness
Threshold networks are often plagued with random and statistical noise at low thresholds, which can hinder their accuracy.As the threshold increases, the network becomes sparser by removing spurious and weak edges.However, this deletion of edges with smaller weights, or correlation coefficients, comes at the cost of information loss.Conversely, a smaller threshold value results in a larger number of random and spurious edges.To address these challenges and determine the optimal threshold value that minimizes spurious connections while retaining maximum information, we employ the random matrix theory (RMT).Random matrix theory, extensively used in diverse fields such as the study of RNA structures 30,31 , proteins 13 , stock market 32 , wireless communication 33 and many more, provides a framework for analyzing correlation matrices and filtering out noise.Specifically, we utilize a class of matrices called Wishart matrices, which have been extensively used for information filtering of correlation matrices to separate the relevant information from the noise 13,32,34 .We used Wishart matrices to establish the lower bounds for random thresholds.By numerically simulating the bounds of correlation coefficients using an ensemble of Wishart matrices (with dimensions the same as the system), we obtain a reliable approximation for the lower threshold that effectively reduces noise.To validate the effectiveness of our approach, we compare the lower bounds obtained from Wishart matrices with those obtained by shuffling the original data.Remarkably, both methods yield nearly identical results, affirming the reliability of our findings.
For our dataset, the bounds of average random correlations C rand ij are given by −0.2 < C rand ij < 0.2 .Accord- ingly, we classify the threshold values into three distinct regions.The first region corresponds to θ < 0.2 , where randomness dominates, making it challenging to determine genuine interactions amidst spurious ones.As the (2) threshold value surpasses 0.2 ( θ > 0.2 ), we enter the second region, where noise progressively decreases, allowing the emergence of actual links.Notably, a small band exists just beyond the 0.2 threshold, indicating that randomness still contributes significantly to the network links.To achieve a significant reduction in noise, we find that a threshold value of θ > 0.3 is effective for the system.At higher thresholds, we observe links that reflect strong co-evolution between positions during the course of evolution.It is worth noting that many of these links are crucial for the enzymatic activity and structural stability of the protein family.

Properties of individual layer
We conducted an extensive analysis of the topological properties of each individual layer at varying thresholds, with detailed results presented in Table .1.It is crucial to emphasize that these topological properties are influenced by the specific layer being analyzed as well as the chosen threshold values.In the context of the protein threshold network, a node's degree signifies the number of evolutionary interactions that a position maintains with other positions within the protein sequence.Meanwhile, the threshold determines the minimum required strength for these interactions to be considered valid.Nodes with higher degrees represent a highly interacting position.When observing the variation of degree with thresholds, we noted that below the 0.25 threshold, all nodes exhibited significantly high degrees, primarily due to statistical noise.In this region, it is difficult to separate random noise from the system information.However, randomness decreases with the increase in threshold, and at higher thresholds, the nodes with non-zero degrees emerge as vital contributors to the enzyme's functionality.At threshold 0.8, only a select few nodes ( 38, 41, 97, 98, 99, 200, 202, 211) show interaction in at least one layer of the multiplex network.These nodes correspond with the following Ambler numbers (AN) 70, 73, 130, 131, 132, 234, 236, 247 [All Ambler number (AN) will be given in boldface].A comprehensive mapping between the nodes in the network and the Ambler number is provided in the Supplementary Material.Among these, position 38 serves as the primary catalytic residue, while positions 41, 97, 98, 99, 200, and 202 are also associated with catalytic functions 35,36 .All these positions are present in the hydrophobicity layer, except position 211 appears in polarizability found to be involved in the acylation mechanism 37 .
In our analysis, we have identified that a limited number of positions participate in evolutionary interactions within the network.For instance, at a threshold of 0.5, only 103 distinct positions (out of 248 positions) actively contribute in at least one layer of the multiplex network, with 145 positions either lacking interactions or displaying interaction strengths below 0.5.We identified 57 positions associated with hydrophobicity, 46 positions with polarizability, 44 positions with volume, and 53 positions with polarity, all exhibiting non-zero   36,38 , position 211 was found to be involved in the acylation mechanism 37 while other positions 92 and 177 are the conserved residues 38 , position 187 was found to have interaction with ceftazidime and cefotaxime 36 .Furthermore, increasing the threshold, led to the elimination of most of the nodes, retaining only those with strong evolutionary interactions.As discussed earlier, at 0.8 threshold only 8 out of 248 positions have non-zero interactions.These positions, especially in the hydrophobicity layer, coincide with a conserved motif for the protein family 35,36 , highlighting their crucial role in controlling the protein's activity, often linked to catalytic or substrate binding positions 35,36 .At higher thresholds, nodes with high clustering coefficients form the core of important motifs within the enzyme family, particularly within the hydrophobicity layer, where a modular structure emerges due to collaborative functional interdependence during the evolutionary process.Physically, in the protein network, the nodes with high clustering coefficients indicate that they are surrounded by positions that have strong interactions with each other.Therefore, these nodes are part of the neighborhood that has highly interconnected neighbors with strong interaction, indicating a high possibility of functional and evolutionary constraints.These neighborhoods are sometimes part of the sector that forms the important regions in the proteins.For example, at 0.  3].Some of these positions are catalytic and/or involved in substrate binding/activity 97, 98, 101, 103, 111, 123, 124, 199, 200, 201, 203 35,36 whereas others are highly conserved positions 46, 90, 110, 149, 150, 157, 182, 209 35 , positions 208 is conserved as well as involved in stabilization of the initial enzyme-substrate complex 37 .Position 211 was found to be involved in the acylation mechanism 37 .However, four positions, namely 105, 115, 160, and 194, lack existing literature references for the class-A β -lactamase family.Interestingly, positions 105, 115, and 160 predominantly appear in the polarity layer, indicating their interactions involve polar forces, while position 194 seems to engage in hydrophobic interactions.These positions with high clustering coefficients are part of a neighborhood with extensive evolutionary connections.This suggests that they could hold significance within the family and might serve as promising targets for drug development.It is important to note that the observed clustering patterns are specific to each property, allowing for the identification of evolutionary interactions between positions for a specific property.
The presence of the modular structure in the network may be the outcome of the collaborative functional interdependence during the evolutionary process.The components of the network extracted at different thresholds have both structural and functional significance.
The class-A β-lactamase family contains four conserved motifs which are 38 SXXK 41 , 97 SDN 99 , 133 EXXLN 137 and 200 KTG 202 [AN: 70 SXXK 73 , 130 SDN 132 , 166 EXXLN 170 and 234 KTG 236 ] The graph-theoretical approach uniquely and distinctly extracts these motifs based on the hydrophobicity scale (at θ = 0.8), which also gives the interac- tion between positions.The strongest component 200 − 202 is extracted at threshold 0.95 corresponds to the Table 1.Topological Properties of the class-A β-lactamase family such as density, number of edges, average clustering ( C avg ), average degree ( K avg ), maximum degree ( K max ), size of largest component ( N comp ), average path length ( L avg ), Radius (R), and Average Eccentricity ( ε ) at different threshold θ. .Two positions 97 and 99 of the third motif 97 SDN 99 appear at the 0.9 threshold (appears in Polarizability and Volume at 0.85 threshold ) which becomes the complete SDN motif at 0.85 with the addition of 98.This implies that the sites 97 (S) and 99 (N) have strong interactions between them while the interactions with position 98 (D) are not as strong.There are only three components 38 − 41 , 200 − 202 , and 97 − 98 − 99 in the threshold range 0.9 to 0.78.The robustness of these components against varying thresholds gives the superiority of these positions and motifs over other positions.Most of these positions are the major constituent of the specificity-determining sites that contribute to the active, catalytic, and ligand binding sites.
The decrease in threshold to 0.75 results in the addition of new components, along with 133 − 137 identified as the 133 EXXLN 137 motif.Although there is considerable overlap between interaction patterns (and the positions with non-zero links) at low thresholds for different properties but there is a significant divergence at intermediate and high thresholds, indicating unique information revealed by each property.Furthermore, the patterns observed are property-dependent, allowing for the identification of evolutionary interactions between positions for a specific property.The components are shown in Fig. 2 and contributing nodes are given in Supplementary Material Table 3.Most of the positions are crucial for the family as shown by literature studies [35][36][37][38] , and many of these positions may hold the potential for acting as targets for future inhibitor design and antibiotic development.

Multi-link and multi-adjacency
In a multiplex network, we can define multi-links 15,16 by considering a vector � m = (m 1 , m 2 , . . ., m α , . . ., m L ) where each element m α can takes either either the value 0 or 1.A multi-link between two nodes is defined by the vector m such that m α = 1 if the two nodes are connected by a link in layer α and zero otherwise.In general, a multi-link between two nodes say i and j is given using the layer adjacency matrix as , where a α being the adjacency matrix for network layer α .If two given nodes are connected in every layer then � m = � 1 whereas multi-link � m = � 0 signifies that the nodes are not directly connected in any layer of the multiplex network.For L layers there are 2 L possible multi-links.
Using multi-links, we define the multi-Adjacency matrix 15,16 as A m with the element ⊣ m ij as 1, when there exists a multi-link m between node i and j and 0 other wise.Mathematically, the element A m ij of multi-Adjacency matrix are given in terms of the adjacency matrices a α of layers as Due to the normalization condition � m A � m ij = 1 , only 2 L−1 out of total of 2 L adjacency matrix (one for each m ) are independent.The normalization condition � m A � m ij = 1 implies that if a pair of nodes have one type of multi-link it cannot have another multi-link.The total number of multi-links in a network is equal to the total number of interactions among all pairs of nodes in the network.With multi-adjacency matrix, one can define the multi-degree K m i of a node i as Multi-degree of a node represents the number of multi-links connected to that particular node.For example, a network with only two layers ( L = 2 ), the degree This value indicates the total number of nodes that are simultaneously connected to node i in both layers 1 and layer 2.
In the context of the class-A β-lactamase multiplex network, there are four layers arranged in order of sig- nificance: polarity (0001), volume (0010), polarizability (0100), and hydrophobicity (1000).When referring to a specific multi-degree value, such as 1000, it indicates links that are exclusively present in the hydrophobicity layer.On the other hand, a multi-degree of 1001 suggests links that are common to both the hydrophobicity and polarity layers.The multi-degree of each node at the different thresholds for class-A β-lactamase family is shown in Supplementary Material Figures 5-7.For the class-A β-lactamase family, at 0.1 thresholds, many pairs of nodes are connected across all network layers simultaneously.These nodes exhibit a non-zero multi-degree value, denoted as K m i , where � m = (1, 1, 1, 1) .Out of the total 248 positions, 233 positions have non-zero values for the multi-degree Ki 1111 .The presence of a non-zero multidegree K 1111 i for a considerable number of nodes at low threshold values can be attributed to statistical noise.With an increase in the threshold, the noise in the system becomes filtered, resulting in fewer pairs of connected nodes in each layer.For instance, at 0.5 threshold, there are only five nodes 46, 124, 137, 156, 199 [AN: 78, 157, 170, 189, 233] with a non-zero multidegree K (1111) i .This indicates that these positions interact in every layer of the network.Notably, positions 124, 137, and 199 are critical residues for catalytic action 38,39 , position 46 shows high conservation ( 96% ) for Gram-negative bacteria 38 .Moreover, certain substitutions at position 156 are known to abolish TEM-1 β-lactamase activity 40 .However, at a threshold of 0.6, only two nodes 199 and 124 (both significant catalytic residues 38 ) remain connected in each layer, both having a multidegree of 1, i.e., K (1111) i = 1 , while all other nodes have a multidegree 1 is 0. As we further increase the threshold, no nodes display a non-zero multidegree K (1111) i .In class-A β-lactamase, the omega loop (positions 128-146 with AN: 161-179) plays a crucial role in substrate recognition and catalysis 39 .Mutations in this loop can affect the enzyme's adaptation to new antibiotics.In our Vol.:(0123456789) www.nature.com/scientificreports/analysis at 0.5 threshold, we identified 103 positions that have non-zero multi-degree.Among these, 12 positions fall within the omega loop, that have n0n-zero multi-degree and serve specific functions.Among these, positions 128 and 134 are linked to substrate specificity 36 , Positions 129, 131, 133, 136, 137, and 146 are involved in catalytic and/or substrate binding 38,39 .The remaining positions 130, 142, 143, and 144 are conserved across class-A β-lactamases 38 .Position 134 which influences substrate activity is active mainly in the polarity layer whereas position 128 interacts more in the polarizability layer.The other catalytic and conserved positions show higher contributions in the hydrophobicity layer or its combination with other properties [Supplementary Material Figures 5-7].If we further lower the threshold to 0.4, all positions in the omega loop have nonzero multi-degree, except for three positions 132, 138, and 140 [Supplementary Material Table 3].
The quantity K m i serves as a local measure of the link overlap between layers in the multiplex network.It also indicates the strength of interaction between positions within each property layer.For instance, if a position exhibits higher values of , it signifies a greater involvement of that position in hydrophobic interactions rather than polar interactions.The multidegree can establish a hierarchy in the influence of physiochemical properties at a given position.One of the conserved motifs of class-A β-lactamase family, 97 SDN 99 [Supplementary Material Figures 5-7], positions 97, 98, and 99 exhibit varying multidegrees when considering different physiochemical properties represented as distinct layers in the multiplex network analysis.In the hydrophobicity layer (layer 1000), positions 97, 98, and 99 display multidegrees of 1, 1, and 2, respectively.However, when examining the polarizability layer (layer 0100), only position 97 demonstrates a multidegree of 1, while the other two positions have a multidegree of zero.As for the volume layer (layer 0010) and the polarity layer (layer 0001), none of the positions in the conserved motif exhibit a nonzero multidegree.This observation highlights the dominance of hydrophobicity in influencing the positions and the conserved motif.It further suggests that the other properties, with the exception of hydrophobicity (and polarizability for position 97), do not significantly influence these specific positions.Hence, the multi-layer structure of the multiplex network establishes a hierarchy in terms of the influence exerted by each property on individual positions within the class-A β-lactamase family.
Figure 3 displays the distribution of multi-links in the multiplex network at various thresholds.The analysis reveals interesting patterns among different property layers.At low and intermediate thresholds ( θ < 0.5 ), Polar- ity (0001) consistently exhibits the highest number of unique evolutionary links, indicating exclusive connections between specific nodes in this layer compared to others.It is followed by hydrophobicity (1000) and polarizability (0100) with a substantial number of unique links.Conversely, the volume layer (0010) demonstrates the lowest number of unique links.However, at higher thresholds ( θ > 0.5 ), hydrophobicity surpasses polarity in terms of unique links, suggesting a stronger overall evolutionary interaction within the hydrophobicity compared to polarity.
To investigate the collective behavior of property combinations, we examine the simultaneous connections present in different layer combinations.The combination of polarizability and volume (0110) exhibits the highest number of simultaneous connections present in both layers.This is followed by the combinations of hydrophobicity-polarity (1001) and hydrophobicity-polarizability (1100).There are two possible explanations for this observation.First, the two layers may encode similar information (statistical noise), resulting in a higher similarity between the links present in both layers.Alternatively, the two positions may be evolutionarily connected, constituting a functional or structural motif where the properties play vital roles.In such cases, any evolutionary change in one position would affect both properties simultaneously, while leaving other properties www.nature.com/scientificreports/unaffected.On the other hand, the combinations of layers (0011, 0101, 1010) exhibit almost zero corresponding multidegrees at intermediate and higher thresholds ( θ > 0.4 ).This suggests that these property combinations have little impact on the evolutionary interaction between positions.Remarkably, the combination of hydrophobicity, polarizability, and volume (1110) reveals links that are common to all three layers, but absent in the polarity layer.These connections survive even at a very high threshold of 0.8, indicating evolutionary conservation.The presence of these three properties may have an evolutionary impact on protein sequences by potentially imposing functional or structural constraints.In contrast, the other combinations of three properties (0111, 1011, and 1101) exhibit considerably fewer links compared to 1110 at intermediate and higher thresholds.The combination of properties in 1110 (hydrophobicity, polarizability, and volume ) is particularly crucial for the β-lactamase family.At higher thresholds ( θ > 0.4 ) several key positions stand out by showing a high value of multi-degree.For instance at 0.6 threshold for the hydrophobicity layer (1000), the identified positions 97 and 200 play role in substrate binding and catalytic process 38 , position 123, 124, 146, 149, are catalytic positions 38 and position 208 has an impact cephalosporin resistance 37 .In the polarity layer (0001), the position 187 interacts with ceftazidime and cefotaxime 36 along with positions 157, 165, 173, and 177 being conserved 38 .Our analysis also uncovers that distinct properties and their combinations highlight different interacting positions.These properties, organized as layers, exhibit varying degrees of influence on evolutionary interactions within the multiplex network.We find that the majority of interacting positions are in the hydrophobicity layer, followed by polarity, polarizability, and, finally, volume (Figures 5-7 in Supplementary Material).This observation implies that hydrophobicity has the most substantial influence on evolutionary interactions, while volume has the least impact.This hierarchy provides a ranking of physicochemical properties in determining evolutionary interactions and positions within the family.The multi-layer structure of the multiplex network provides valuable insights into the hierarchy of physiochemical properties influencing individual positions within the class-A β-lactamase family, facilitating a targeted approach for understanding and manipulating enzymatic activity to combat antibiotic resistance.This hierarchical ranking of the properties is possible by considering the multiplex structure of the enzyme family.
For any two layers ( α and γ ) of the network, we can define also a global measure of overlap called link overlap, which measures the ratio of the common links in both layers.Link overlap quantifies the links connecting the same pair of nodes in both layers and is defined as The link overlap between 4 layers of the multiplex network is shown in Fig. 4. At a low threshold ( θ = 0.1 ), there is a significant overlap in the interaction among all properties.Specifically, polarizability (represented by 2) and volume (represented by 3) exhibit a remarkably high overlap of over 70%.As the threshold increases, the overlap between layers in the network starts to decrease, although polarizability and volume continue to show significant overlap.As discussed previously, the threshold range 0.0 − 0.2 represents a noisy region mostly plagued by randomness.The high overlap in this region can be attributed to the random links between positions but these links are very weak in terms of their strength.In the threshold region where the information is predominant ( θ > 0.3 ), there is a sudden decrease in the overlaps and only the actual link contributes.For example, the overlap between polarity (property 4), with volume and polarizability is over 25% at threshold 0.1 but reduces to almost zero at threshold 0.7.Volume and polarizability consistently exhibit an high overlap of approximately 60% across all thresholds, indicating that they encode similar information.Consequently, it will be reasonably safe to reduce the size of the multiplex network by eliminating one of the two layers (either volume or polarizability) with minimal loss of information.

Multi-strength
Each network layer G α can be independently represented using an adjacency matrix a α for physiochemical prop- erty α with a α ij = 1 , if E α i,j > 0 and 0 otherwise.Additionally, a weight matrix w α can be defined for property α with w α ij = E α i,j , if nodes i and j are connected and zero otherwise.In weighted networks, to check the heterogeneity in www.nature.com/scientificreports/ the distribution of weights across the edges, local parameters such as strength are used 15 .The strength of a node i in layer α is defined as S α i = N j=1 w α ij , which is the sum of weights of all the links incident upon that node.Figure 5 reveals important insights about the influence strength of positions in the multiplex network layers.At a low threshold ( θ = 0.3 ), almost all positions exhibit non-zero influence strength.However, as the thresh- old increases, only a subset of positions retains non-zero influence strength, indicating their significance in the network.Each multiplex layer exhibits distinct nodes with significant influence strength, and these differences become more pronounced at higher thresholds.

Conclusions
The use of a multiplex network to analyze the evolutionary interactions between protein sequences yields unparalleled insights into the structural and functional motifs of the class-A β-lactamase family.By deriving each layer of the network from a correlation matrix calculated using physiochemical properties, we unveil novel information about the intricate interactions between nodes, allowing us to selectively determine key positions and interactions that may act as potential targets for influencing enzymatic or catalytic activity.Although each layer is derived from the same MSA, but it unravels a piece of different information in terms of interaction between the nodes giving useful insight into the functionality and structure of the protein family.We also observe that interaction between the positions depends on the physiochemical properties where positions tend to cluster into groups with identical signs of the property.Our methodology also reveals the hierarchy in the influence of physiochemical properties at a given position, pinpointing the most relevant property responsible for the protein's functionality.Link overlap analysis reveals that there are limited information exchanges between any two layers, indicating the importance of combining layers to shed light on their collective behavior.The combination of hydrophobicity, polarizability, and volume exhibits common links across all layers, suggesting functional or structural constraints for the class-A β-lactamase family.In conclusion, the application of a multiplex network provides valuable insights into the function and structure of the class-A β-lactamase protein family, in identifying key positions, interactions as well as combinations of physiochemical properties.Furthermore, the multiplex network proposed in this study exhibits considerable potential for broader utilization across various protein families, offering invaluable insights into their structural and functional characteristics.This novel technique represents a potent approach for investigating the impact of diverse properties on protein functionality, effectively elucidating crucial motifs and unveiling the intricate hierarchical connections between properties and evolutionary constraints that govern these protein families.

Figure 2 .
Figure 2. Layers of the multiplex network at different thresholds (cut-off) on correlation coefficients [0.3 (a-d), 0.5 (e-h), 0.7 (i-l)] showing only nodes with non-zero connections for clarity.The blue color indicates a negative average value of the physiochemical property at that position whereas the red color represents the positive average value.Node numbering follows the filtered MSA, differing from the Ambler numbering scheme.

Figure 3 .
Figure 3. Distribution of multi-links in the network at the different thresholds.The properties are represented from least significant bit to most significant in order of Polarity, Volume, Polarizability, and hydrophobicity.

Figure 4 .
Figure 4. Link overlap between different layers of the multiplex network at the different thresholds.The layers display the physiochemical properties in order 1-hydrophobicity, 2-polarizability, 3-volume, and 4-polarity.